If there exists a homotopy equivalence between XX and YY we say that XX and YY are homotopy equivalent or that they have the same homotopy type. Being homotopy equivalent is evidently an equivalence relation.

Strong homotopy equivalences

Sometimes an apparently stronger form of homotopy equivalence is needed. These seem to have been first noticed by Richard Lashof in 1970 in work on triangulation and smoothing. The point is one of the interaction between the two homotopies, which are not made explicit in the definition above. Let us set this up slightly differently:

In a homotopy equivalence there are the two maps f:X→Yf: X\to Y and g:Y→Xg: Y\to X, then two homotopies

H:X×I→X,H:g∘f∼1XH : X\times I \to X, H:g \circ f \sim 1_X

and

K:Y×I→Y,K:f∘g∼1Y.K: Y\times I \to Y, K : f \circ g \sim 1_Y.

The whiskering actions of maps on homotopies (and more generally in any (∞,1)-category or category with a cylinder functor) gives two homotopies f∘g∘f∼ff\circ g \circ f \sim f, namely f*H=f∘H:X×I→Yf_*H = f\circ H : X\times I \to Y and f*K=K∘(f×I)f^*K = K\circ (f\times I). Similarly, of course, there are two homotopies g∘f∘g∼gg\circ f \circ g \sim g, namely g*Kg_*K and g*Hg^*H. In the usual definition of homotopy equivalence, there is no coherence required between these. That is handled precisely by the notion of stong homotopy equivalence. More precisely Lashof defined

Definition

A strong homotopy equivalence between spaces XX and YY is a quadruple (f,g,H,K)(f,g,H,K), as above, such that f*H∼f*Kf_*H \sim f^*K and g*K∼g*Hg_*K\sim g^*H.

Thus this imposes a minimal coherence condition on the data making up the homotopy equivalence. The definition clearly can be generalised to any reasonable setting with a notion of homotopy.

The question naturally arises as to whether all homotopy equivalences are strong. Rainer Vogt(1972) proved

Vogt’s Lemma

If f:X→Yf: X\to Y be a morphism that is a homotopy equivalence in TopTop, let g:Y→Xg: Y\to X be a homotopy inverse and H:g∘f∼1XH:g \circ f \sim 1_X a homotopy. Then there is a homotopy K:f∘g∼1YK: f \circ g \sim 1_Y such that (f,g,H,K)(f,g,H,K) is a strong homotopy equivalence.

Various versions of this are known in other settings, e.g. SSetSSet-enriched categories. In a 2-category the corresponding fact is that any equivalence can be improved to an adjoint equivalence, by changing at most one of the 2-cell isomorphisms involved.

Note, though, that the coherence supplied by Vogt’s lemma is not required to continue to higher levels. There is no condition of compatibilty between the two ‘homotopies between the homotopies’. In some situations, at least, some higher compatibility is known to be derivable; for instance, any biequivalence in a 3-category can be improved to an adjoint biequivalence.

Tim: I do not know if there is a neat formulation of the full homotopy coherent version of this, nor exactly in what settings the analogous abstract versions of Vogt’s lemma go across

Mike Shulman: I think there’s a “full coherentification” version for quasicategories in HTT somewhere.